Mathematical Basis behind inflation of Standard errors of Regression estimates due to multicollinearity

We know that due to multi-collinearity, the standard errors of beta estimates get inflated. But what is the mathematical basis to it?

I am looking for some mathematical relationship or something to explain this.

Like I understand if standard error of betas goes up, then t-statistics goes down and we might not be able to reject the null or the variables would appear non-significant.

But what is the mathematical relationship between multicolllinearity and inflation in variance of coefficients?

• When some linear combinations of predictors (including the constant) are almost zero, the variance of a coefficient for that combination will have extremely large variance (there's little-to-no information in the data about it). That "projects" onto the coefficients in your model ... every variable that appears in that poorly-determined linear combination will 'inherit' some of that indeterminacy (i.e. get a large variance for its estimated coefficient). May 3, 2015 at 3:16

He gives a formula for the estimated variance of OLS regressor $\beta_k$ in a regression of $y$ on $K$ variables as $$\frac{1}{N-K}\cdot\frac{\sigma^2_y}{\sigma^2_k}\cdot\frac{1-R^2}{1-R^2_k},$$ where $\sigma^2_y$ is the estimated variance of $y$, $\sigma^2_k$ is the estimated variance of $x_k$, $R_k^2$ is from the regression of $x_k$ on $K-1$ remaining independent variables, and $N$ is the sample size. The set of $K$ already includes a constant.
As the independent variables get more collinear, $R^2_k$ approaches one, so the variance blows up.
• The specific value $1/(1-R^2_k)$ is called the variance inflation factor. It's 1 only when $x_k$ is orthogonal to all other predictors in the model. Nov 21, 2018 at 16:01